Theorem sibfof 30402

Description: Applying function operations on simple functions results in simple functions with regard to the destination space, provided the operation fulfills a simple condition. (Contributed by Thierry Arnoux, 12-Mar-2018.)

Hypotheses

Ref	Expression
sitgval.b	|- B = ( Base ` W )
sitgval.j	|- J = ( TopOpen ` W )
sitgval.s	|- S = (sigaGen ` J )
sitgval.0	|- .0. = ( 0g ` W )
sitgval.x	|- .x. = ( .s ` W )
sitgval.h	|- H = (RRHom ` (Scalar ` W ) )
sitgval.1	|- ( ph -> W e. V )
sitgval.2	|- ( ph -> M e. U. ran measures )
sibfmbl.1	|- ( ph -> F e. dom ( Wsitg M ) )
sibfof.c	|- C = ( Base ` K )
sibfof.0	|- ( ph -> W e. TopSp )
sibfof.1	|- ( ph -> .+ : ( B X. B ) --> C )
sibfof.2	|- ( ph -> G e. dom ( Wsitg M ) )
sibfof.3	|- ( ph -> K e. TopSp )
sibfof.4	|- ( ph -> J e. Fre )
sibfof.5	|- ( ph -> ( .0. .+ .0. ) = ( 0g ` K ) )

Assertion

Ref	Expression
sibfof	|- ( ph -> ( F oF .+ G ) e. dom ( Ksitg M ) )

Proof of Theorem sibfof

Dummy variables x y z p b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sibfof.1	. . . . . . . 8 |- ( ph -> .+ : ( B X. B ) --> C )
2		sibfof.0	. . . . . . . . . . 11 |- ( ph -> W e. TopSp )
3		sitgval.b	. . . . . . . . . . . 12 |- B = ( Base ` W )
4		sitgval.j	. . . . . . . . . . . 12 |- J = ( TopOpen ` W )
5	3, 4	tpsuni 20740	. . . . . . . . . . 11 |- ( W e. TopSp -> B = U. J )
6	2, 5	syl 17	. . . . . . . . . 10 |- ( ph -> B = U. J )
7	6	sqxpeqd 5141	. . . . . . . . 9 |- ( ph -> ( B X. B ) = ( U. J X. U. J ) )
8	7	feq2d 6031	. . . . . . . 8 |- ( ph -> ( .+ : ( B X. B ) --> C <-> .+ : ( U. J X. U. J ) --> C ) )
9	1, 8	mpbid 222	. . . . . . 7 |- ( ph -> .+ : ( U. J X. U. J ) --> C )
10	9	fovrnda 6805	. . . . . 6 |- ( ( ph /\ ( z e. U. J /\ x e. U. J ) ) -> ( z .+ x ) e. C )
11		sitgval.s	. . . . . . 7 |- S = (sigaGen ` J )
12		sitgval.0	. . . . . . 7 |- .0. = ( 0g ` W )
13		sitgval.x	. . . . . . 7 |- .x. = ( .s ` W )
14		sitgval.h	. . . . . . 7 |- H = (RRHom ` (Scalar ` W ) )
15		sitgval.1	. . . . . . 7 |- ( ph -> W e. V )
16		sitgval.2	. . . . . . 7 |- ( ph -> M e. U. ran measures )
17		sibfmbl.1	. . . . . . 7 |- ( ph -> F e. dom ( Wsitg M ) )
18	3, 4, 11, 12, 13, 14, 15, 16, 17	sibff 30398	. . . . . 6 |- ( ph -> F : U. dom M --> U. J )
19		sibfof.2	. . . . . . 7 |- ( ph -> G e. dom ( Wsitg M ) )
20	3, 4, 11, 12, 13, 14, 15, 16, 19	sibff 30398	. . . . . 6 |- ( ph -> G : U. dom M --> U. J )
21		dmexg 7097	. . . . . . 7 |- ( M e. U. ran measures -> dom M e. _V )
22		uniexg 6955	. . . . . . 7 |- ( dom M e. _V -> U. dom M e. _V )
23	16, 21, 22	3syl 18	. . . . . 6 |- ( ph -> U. dom M e. _V )
24		inidm 3822	. . . . . 6 |- ( U. dom M i^i U. dom M ) = U. dom M
25	10, 18, 20, 23, 23, 24	off 6912	. . . . 5 |- ( ph -> ( F oF .+ G ) : U. dom M --> C )
26		sibfof.3	. . . . . . . 8 |- ( ph -> K e. TopSp )
27		sibfof.c	. . . . . . . . 9 |- C = ( Base ` K )
28		eqid 2622	. . . . . . . . 9 |- ( TopOpen ` K ) = ( TopOpen ` K )
29	27, 28	tpsuni 20740	. . . . . . . 8 |- ( K e. TopSp -> C = U. ( TopOpen ` K ) )
30	26, 29	syl 17	. . . . . . 7 |- ( ph -> C = U. ( TopOpen ` K ) )
31		fvex 6201	. . . . . . . 8 |- ( TopOpen ` K ) e. _V
32		unisg 30206	. . . . . . . 8 |- ( ( TopOpen ` K ) e. _V -> U. (sigaGen ` ( TopOpen ` K ) ) = U. ( TopOpen ` K ) )
33	31, 32	ax-mp 5	. . . . . . 7 |- U. (sigaGen ` ( TopOpen ` K ) ) = U. ( TopOpen ` K )
34	30, 33	syl6eqr 2674	. . . . . 6 |- ( ph -> C = U. (sigaGen ` ( TopOpen ` K ) ) )
35	34	feq3d 6032	. . . . 5 |- ( ph -> ( ( F oF .+ G ) : U. dom M --> C <-> ( F oF .+ G ) : U. dom M --> U. (sigaGen ` ( TopOpen ` K ) ) ) )
36	25, 35	mpbid 222	. . . 4 |- ( ph -> ( F oF .+ G ) : U. dom M --> U. (sigaGen ` ( TopOpen ` K ) ) )
37	31	a1i 11	. . . . . . 7 |- ( ph -> ( TopOpen ` K ) e. _V )
38	37	sgsiga 30205	. . . . . 6 |- ( ph -> (sigaGen ` ( TopOpen ` K ) ) e. U. ran sigAlgebra )
39		uniexg 6955	. . . . . 6 |- ( (sigaGen ` ( TopOpen ` K ) ) e. U. ran sigAlgebra -> U. (sigaGen ` ( TopOpen ` K ) ) e. _V )
40	38, 39	syl 17	. . . . 5 |- ( ph -> U. (sigaGen ` ( TopOpen ` K ) ) e. _V )
41	40, 23	elmapd 7871	. . . 4 |- ( ph -> ( ( F oF .+ G ) e. ( U. (sigaGen ` ( TopOpen ` K ) ) ^m U. dom M ) <-> ( F oF .+ G ) : U. dom M --> U. (sigaGen ` ( TopOpen ` K ) ) ) )
42	36, 41	mpbird 247	. . 3 |- ( ph -> ( F oF .+ G ) e. ( U. (sigaGen ` ( TopOpen ` K ) ) ^m U. dom M ) )
43		inundif 4046	. . . . . . 7 |- ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) = b
44	43	imaeq2i 5464	. . . . . 6 |- ( `' ( F oF .+ G ) " ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) ) = ( `' ( F oF .+ G ) " b )
45		ffun 6048	. . . . . . . 8 |- ( ( F oF .+ G ) : U. dom M --> C -> Fun ( F oF .+ G ) )
46		unpreima 6341	. . . . . . . 8 |- ( Fun ( F oF .+ G ) -> ( `' ( F oF .+ G ) " ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) ) = ( ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) u. ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) ) )
47	25, 45, 46	3syl 18	. . . . . . 7 |- ( ph -> ( `' ( F oF .+ G ) " ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) ) = ( ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) u. ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) ) )
48	47	adantr 481	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " ( ( b i^i ran ( F oF .+ G ) ) u. ( b \ ran ( F oF .+ G ) ) ) ) = ( ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) u. ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) ) )
49	44, 48	syl5eqr 2670	. . . . 5 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " b ) = ( ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) u. ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) ) )
50		dmmeas 30264	. . . . . . . 8 |- ( M e. U. ran measures -> dom M e. U. ran sigAlgebra )
51	16, 50	syl 17	. . . . . . 7 |- ( ph -> dom M e. U. ran sigAlgebra )
52	51	adantr 481	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> dom M e. U. ran sigAlgebra )
53		imaiun 6503	. . . . . . . 8 |- ( `' ( F oF .+ G ) " U_ z e. ( b i^i ran ( F oF .+ G ) ) { z } ) = U_ z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } )
54		iunid 4575	. . . . . . . . 9 |- U_ z e. ( b i^i ran ( F oF .+ G ) ) { z } = ( b i^i ran ( F oF .+ G ) )
55	54	imaeq2i 5464	. . . . . . . 8 |- ( `' ( F oF .+ G ) " U_ z e. ( b i^i ran ( F oF .+ G ) ) { z } ) = ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) )
56	53, 55	eqtr3i 2646	. . . . . . 7 |- U_ z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) = ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) )
57		inss2 3834	. . . . . . . . . 10 |- ( b i^i ran ( F oF .+ G ) ) C_ ran ( F oF .+ G )
58	6	feq3d 6032	. . . . . . . . . . . . . . 15 |- ( ph -> ( F : U. dom M --> B <-> F : U. dom M --> U. J ) )
59	18, 58	mpbird 247	. . . . . . . . . . . . . 14 |- ( ph -> F : U. dom M --> B )
60	6	feq3d 6032	. . . . . . . . . . . . . . 15 |- ( ph -> ( G : U. dom M --> B <-> G : U. dom M --> U. J ) )
61	20, 60	mpbird 247	. . . . . . . . . . . . . 14 |- ( ph -> G : U. dom M --> B )
62		ffn 6045	. . . . . . . . . . . . . . 15 |- ( .+ : ( B X. B ) --> C -> .+ Fn ( B X. B ) )
63	1, 62	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> .+ Fn ( B X. B ) )
64	59, 61, 23, 63	ofpreima2 29466	. . . . . . . . . . . . 13 |- ( ph -> ( `' ( F oF .+ G ) " { z } ) = U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
65	64	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( `' ( F oF .+ G ) " { z } ) = U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
66	51	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> dom M e. U. ran sigAlgebra )
67	51	ad2antrr 762	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> dom M e. U. ran sigAlgebra )
68		simpll 790	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ph )
69		inss1 3833	. . . . . . . . . . . . . . . . . 18 |- ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( `' .+ " { z } )
70		cnvimass 5485	. . . . . . . . . . . . . . . . . . . 20 |- ( `' .+ " { z } ) C_ dom .+
71		fdm 6051	. . . . . . . . . . . . . . . . . . . . 21 |- ( .+ : ( B X. B ) --> C -> dom .+ = ( B X. B ) )
72	1, 71	syl 17	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> dom .+ = ( B X. B ) )
73	70, 72	syl5sseq 3653	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( `' .+ " { z } ) C_ ( B X. B ) )
74	73	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( `' .+ " { z } ) C_ ( B X. B ) )
75	69, 74	syl5ss 3614	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( B X. B ) )
76	75	sselda 3603	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> p e. ( B X. B ) )
77	51	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> dom M e. U. ran sigAlgebra )
78		sibfof.4	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> J e. Fre )
79	78	sgsiga 30205	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> (sigaGen ` J ) e. U. ran sigAlgebra )
80	11, 79	syl5eqel 2705	. . . . . . . . . . . . . . . . . 18 |- ( ph -> S e. U. ran sigAlgebra )
81	80	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> S e. U. ran sigAlgebra )
82	3, 4, 11, 12, 13, 14, 15, 16, 17	sibfmbl 30397	. . . . . . . . . . . . . . . . . 18 |- ( ph -> F e. ( dom MMblFnM S ) )
83	82	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> F e. ( dom MMblFnM S ) )
84	4	tpstop 20741	. . . . . . . . . . . . . . . . . . . . 21 |- ( W e. TopSp -> J e. Top )
85		cldssbrsiga 30250	. . . . . . . . . . . . . . . . . . . . 21 |- ( J e. Top -> ( Clsd ` J ) C_ (sigaGen ` J ) )
86	2, 84, 85	3syl 18	. . . . . . . . . . . . . . . . . . . 20 |- ( ph -> ( Clsd ` J ) C_ (sigaGen ` J ) )
87	86	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> ( Clsd ` J ) C_ (sigaGen ` J ) )
88	78	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> J e. Fre )
89		xp1st 7198	. . . . . . . . . . . . . . . . . . . . . 22 |- ( p e. ( B X. B ) -> ( 1st ` p ) e. B )
90	89	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 1st ` p ) e. B )
91	6	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> B = U. J )
92	90, 91	eleqtrd 2703	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 1st ` p ) e. U. J )
93		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 |- U. J = U. J
94	93	t1sncld 21130	. . . . . . . . . . . . . . . . . . . 20 |- ( ( J e. Fre /\ ( 1st ` p ) e. U. J ) -> { ( 1st ` p ) } e. ( Clsd ` J ) )
95	88, 92, 94	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. ( Clsd ` J ) )
96	87, 95	sseldd 3604	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. (sigaGen ` J ) )
97	96, 11	syl6eleqr 2712	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 1st ` p ) } e. S )
98	77, 81, 83, 97	mbfmcnvima 30319	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ p e. ( B X. B ) ) -> ( `' F " { ( 1st ` p ) } ) e. dom M )
99	68, 76, 98	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( `' F " { ( 1st ` p ) } ) e. dom M )
100	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfmbl 30397	. . . . . . . . . . . . . . . . . 18 |- ( ph -> G e. ( dom MMblFnM S ) )
101	100	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> G e. ( dom MMblFnM S ) )
102		xp2nd 7199	. . . . . . . . . . . . . . . . . . . . . 22 |- ( p e. ( B X. B ) -> ( 2nd ` p ) e. B )
103	102	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 2nd ` p ) e. B )
104	103, 91	eleqtrd 2703	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ p e. ( B X. B ) ) -> ( 2nd ` p ) e. U. J )
105	93	t1sncld 21130	. . . . . . . . . . . . . . . . . . . 20 |- ( ( J e. Fre /\ ( 2nd ` p ) e. U. J ) -> { ( 2nd ` p ) } e. ( Clsd ` J ) )
106	88, 104, 105	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. ( Clsd ` J ) )
107	87, 106	sseldd 3604	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. (sigaGen ` J ) )
108	107, 11	syl6eleqr 2712	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ p e. ( B X. B ) ) -> { ( 2nd ` p ) } e. S )
109	77, 81, 101, 108	mbfmcnvima 30319	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ p e. ( B X. B ) ) -> ( `' G " { ( 2nd ` p ) } ) e. dom M )
110	68, 76, 109	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( `' G " { ( 2nd ` p ) } ) e. dom M )
111		inelsiga 30198	. . . . . . . . . . . . . . 15 |- ( ( dom M e. U. ran sigAlgebra /\ ( `' F " { ( 1st ` p ) } ) e. dom M /\ ( `' G " { ( 2nd ` p ) } ) e. dom M ) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M )
112	67, 99, 110, 111	syl3anc 1326	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ z e. ran ( F oF .+ G ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M )
113	112	ralrimiva 2966	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> A. p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M )
114	3, 4, 11, 12, 13, 14, 15, 16, 17	sibfrn 30399	. . . . . . . . . . . . . . . . 17 |- ( ph -> ran F e. Fin )
115	3, 4, 11, 12, 13, 14, 15, 16, 19	sibfrn 30399	. . . . . . . . . . . . . . . . 17 |- ( ph -> ran G e. Fin )
116		xpfi 8231	. . . . . . . . . . . . . . . . 17 |- ( ( ran F e. Fin /\ ran G e. Fin ) -> ( ran F X. ran G ) e. Fin )
117	114, 115, 116	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ran F X. ran G ) e. Fin )
118		inss2 3834	. . . . . . . . . . . . . . . 16 |- ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( ran F X. ran G )
119		ssdomg 8001	. . . . . . . . . . . . . . . 16 |- ( ( ran F X. ran G ) e. Fin -> ( ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( ran F X. ran G ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ ( ran F X. ran G ) ) )
120	117, 118, 119	mpisyl 21	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ ( ran F X. ran G ) )
121		isfinite 8549	. . . . . . . . . . . . . . . . 17 |- ( ( ran F X. ran G ) e. Fin <-> ( ran F X. ran G ) ~< om )
122	121	biimpi 206	. . . . . . . . . . . . . . . 16 |- ( ( ran F X. ran G ) e. Fin -> ( ran F X. ran G ) ~< om )
123		sdomdom 7983	. . . . . . . . . . . . . . . 16 |- ( ( ran F X. ran G ) ~< om -> ( ran F X. ran G ) ~<_ om )
124	117, 122, 123	3syl 18	. . . . . . . . . . . . . . 15 |- ( ph -> ( ran F X. ran G ) ~<_ om )
125		domtr 8009	. . . . . . . . . . . . . . 15 |- ( ( ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ ( ran F X. ran G ) /\ ( ran F X. ran G ) ~<_ om ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
126	120, 124, 125	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
127	126	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
128		nfcv 2764	. . . . . . . . . . . . . 14 |- F/_ p ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) )
129	128	sigaclcuni 30181	. . . . . . . . . . . . 13 |- ( ( dom M e. U. ran sigAlgebra /\ A. p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M /\ ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om ) -> U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M )
130	66, 113, 127, 129	syl3anc 1326	. . . . . . . . . . . 12 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M )
131	65, 130	eqeltrd 2701	. . . . . . . . . . 11 |- ( ( ph /\ z e. ran ( F oF .+ G ) ) -> ( `' ( F oF .+ G ) " { z } ) e. dom M )
132	131	ralrimiva 2966	. . . . . . . . . 10 |- ( ph -> A. z e. ran ( F oF .+ G ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
133		ssralv 3666	. . . . . . . . . 10 |- ( ( b i^i ran ( F oF .+ G ) ) C_ ran ( F oF .+ G ) -> ( A. z e. ran ( F oF .+ G ) ( `' ( F oF .+ G ) " { z } ) e. dom M -> A. z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M ) )
134	57, 132, 133	mpsyl 68	. . . . . . . . 9 |- ( ph -> A. z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
135	134	adantr 481	. . . . . . . 8 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> A. z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
136		ffun 6048	. . . . . . . . . . . . . 14 |- ( .+ : ( B X. B ) --> C -> Fun .+ )
137	1, 136	syl 17	. . . . . . . . . . . . 13 |- ( ph -> Fun .+ )
138		imafi 8259	. . . . . . . . . . . . 13 |- ( ( Fun .+ /\ ( ran F X. ran G ) e. Fin ) -> ( .+ " ( ran F X. ran G ) ) e. Fin )
139	137, 117, 138	syl2anc 693	. . . . . . . . . . . 12 |- ( ph -> ( .+ " ( ran F X. ran G ) ) e. Fin )
140	18, 20, 9, 23	ofrn2 29442	. . . . . . . . . . . 12 |- ( ph -> ran ( F oF .+ G ) C_ ( .+ " ( ran F X. ran G ) ) )
141		ssfi 8180	. . . . . . . . . . . 12 |- ( ( ( .+ " ( ran F X. ran G ) ) e. Fin /\ ran ( F oF .+ G ) C_ ( .+ " ( ran F X. ran G ) ) ) -> ran ( F oF .+ G ) e. Fin )
142	139, 140, 141	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ran ( F oF .+ G ) e. Fin )
143		ssdomg 8001	. . . . . . . . . . 11 |- ( ran ( F oF .+ G ) e. Fin -> ( ( b i^i ran ( F oF .+ G ) ) C_ ran ( F oF .+ G ) -> ( b i^i ran ( F oF .+ G ) ) ~<_ ran ( F oF .+ G ) ) )
144	142, 57, 143	mpisyl 21	. . . . . . . . . 10 |- ( ph -> ( b i^i ran ( F oF .+ G ) ) ~<_ ran ( F oF .+ G ) )
145		isfinite 8549	. . . . . . . . . . . 12 |- ( ran ( F oF .+ G ) e. Fin <-> ran ( F oF .+ G ) ~< om )
146	142, 145	sylib 208	. . . . . . . . . . 11 |- ( ph -> ran ( F oF .+ G ) ~< om )
147		sdomdom 7983	. . . . . . . . . . 11 |- ( ran ( F oF .+ G ) ~< om -> ran ( F oF .+ G ) ~<_ om )
148	146, 147	syl 17	. . . . . . . . . 10 |- ( ph -> ran ( F oF .+ G ) ~<_ om )
149		domtr 8009	. . . . . . . . . 10 |- ( ( ( b i^i ran ( F oF .+ G ) ) ~<_ ran ( F oF .+ G ) /\ ran ( F oF .+ G ) ~<_ om ) -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
150	144, 148, 149	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
151	150	adantr 481	. . . . . . . 8 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( b i^i ran ( F oF .+ G ) ) ~<_ om )
152		nfcv 2764	. . . . . . . . 9 |- F/_ z ( b i^i ran ( F oF .+ G ) )
153	152	sigaclcuni 30181	. . . . . . . 8 |- ( ( dom M e. U. ran sigAlgebra /\ A. z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M /\ ( b i^i ran ( F oF .+ G ) ) ~<_ om ) -> U_ z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
154	52, 135, 151, 153	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> U_ z e. ( b i^i ran ( F oF .+ G ) ) ( `' ( F oF .+ G ) " { z } ) e. dom M )
155	56, 154	syl5eqelr 2706	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) e. dom M )
156		difpreima 6343	. . . . . . . . . 10 |- ( Fun ( F oF .+ G ) -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) = ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) )
157	25, 45, 156	3syl 18	. . . . . . . . 9 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) = ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) )
158		cnvimarndm 5486	. . . . . . . . . . 11 |- ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) = dom ( F oF .+ G )
159	158	difeq2i 3725	. . . . . . . . . 10 |- ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) = ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) )
160		cnvimass 5485	. . . . . . . . . . 11 |- ( `' ( F oF .+ G ) " b ) C_ dom ( F oF .+ G )
161		ssdif0 3942	. . . . . . . . . . 11 |- ( ( `' ( F oF .+ G ) " b ) C_ dom ( F oF .+ G ) <-> ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) ) = (/) )
162	160, 161	mpbi 220	. . . . . . . . . 10 |- ( ( `' ( F oF .+ G ) " b ) \ dom ( F oF .+ G ) ) = (/)
163	159, 162	eqtri 2644	. . . . . . . . 9 |- ( ( `' ( F oF .+ G ) " b ) \ ( `' ( F oF .+ G ) " ran ( F oF .+ G ) ) ) = (/)
164	157, 163	syl6eq 2672	. . . . . . . 8 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) = (/) )
165		0elsiga 30177	. . . . . . . . 9 |- ( dom M e. U. ran sigAlgebra -> (/) e. dom M )
166	16, 50, 165	3syl 18	. . . . . . . 8 |- ( ph -> (/) e. dom M )
167	164, 166	eqeltrd 2701	. . . . . . 7 |- ( ph -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) e. dom M )
168	167	adantr 481	. . . . . 6 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) e. dom M )
169		unelsiga 30197	. . . . . 6 |- ( ( dom M e. U. ran sigAlgebra /\ ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) e. dom M /\ ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) e. dom M ) -> ( ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) u. ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) ) e. dom M )
170	52, 155, 168, 169	syl3anc 1326	. . . . 5 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( ( `' ( F oF .+ G ) " ( b i^i ran ( F oF .+ G ) ) ) u. ( `' ( F oF .+ G ) " ( b \ ran ( F oF .+ G ) ) ) ) e. dom M )
171	49, 170	eqeltrd 2701	. . . 4 |- ( ( ph /\ b e. (sigaGen ` ( TopOpen ` K ) ) ) -> ( `' ( F oF .+ G ) " b ) e. dom M )
172	171	ralrimiva 2966	. . 3 |- ( ph -> A. b e. (sigaGen ` ( TopOpen ` K ) ) ( `' ( F oF .+ G ) " b ) e. dom M )
173	51, 38	ismbfm 30314	. . 3 |- ( ph -> ( ( F oF .+ G ) e. ( dom MMblFnM (sigaGen ` ( TopOpen ` K ) ) ) <-> ( ( F oF .+ G ) e. ( U. (sigaGen ` ( TopOpen ` K ) ) ^m U. dom M ) /\ A. b e. (sigaGen ` ( TopOpen ` K ) ) ( `' ( F oF .+ G ) " b ) e. dom M ) ) )
174	42, 172, 173	mpbir2and 957	. 2 |- ( ph -> ( F oF .+ G ) e. ( dom MMblFnM (sigaGen ` ( TopOpen ` K ) ) ) )
175	64	adantr 481	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( `' ( F oF .+ G ) " { z } ) = U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
176	175	fveq2d 6195	. . . . . 6 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) = ( M ` U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
177		measbasedom 30265	. . . . . . . . 9 |- ( M e. U. ran measures <-> M e. (measures ` dom M ) )
178	16, 177	sylib 208	. . . . . . . 8 |- ( ph -> M e. (measures ` dom M ) )
179	178	adantr 481	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> M e. (measures ` dom M ) )
180		eldifi 3732	. . . . . . . 8 |- ( z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) -> z e. ran ( F oF .+ G ) )
181	180, 113	sylan2 491	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> A. p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M )
182	126	adantr 481	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om )
183		sneq 4187	. . . . . . . . . . 11 |- ( x = ( 1st ` p ) -> { x } = { ( 1st ` p ) } )
184	183	imaeq2d 5466	. . . . . . . . . 10 |- ( x = ( 1st ` p ) -> ( `' F " { x } ) = ( `' F " { ( 1st ` p ) } ) )
185		sneq 4187	. . . . . . . . . . 11 |- ( y = ( 2nd ` p ) -> { y } = { ( 2nd ` p ) } )
186	185	imaeq2d 5466	. . . . . . . . . 10 |- ( y = ( 2nd ` p ) -> ( `' G " { y } ) = ( `' G " { ( 2nd ` p ) } ) )
187		ffun 6048	. . . . . . . . . . . 12 |- ( F : U. dom M --> U. J -> Fun F )
188	18, 187	syl 17	. . . . . . . . . . 11 |- ( ph -> Fun F )
189		sndisj 4644	. . . . . . . . . . 11 |- Disj x e. ran F { x }
190		disjpreima 29397	. . . . . . . . . . 11 |- ( ( Fun F /\ Disj x e. ran F { x } ) -> Disj x e. ran F ( `' F " { x } ) )
191	188, 189, 190	sylancl 694	. . . . . . . . . 10 |- ( ph -> Disj x e. ran F ( `' F " { x } ) )
192		ffun 6048	. . . . . . . . . . . 12 |- ( G : U. dom M --> U. J -> Fun G )
193	20, 192	syl 17	. . . . . . . . . . 11 |- ( ph -> Fun G )
194		sndisj 4644	. . . . . . . . . . 11 |- Disj y e. ran G { y }
195		disjpreima 29397	. . . . . . . . . . 11 |- ( ( Fun G /\ Disj y e. ran G { y } ) -> Disj y e. ran G ( `' G " { y } ) )
196	193, 194, 195	sylancl 694	. . . . . . . . . 10 |- ( ph -> Disj y e. ran G ( `' G " { y } ) )
197	184, 186, 191, 196	disjxpin 29401	. . . . . . . . 9 |- ( ph -> Disj p e. ( ran F X. ran G ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
198		disjss1 4626	. . . . . . . . 9 |- ( ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( ran F X. ran G ) -> (Disj p e. ( ran F X. ran G ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) -> Disj p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
199	118, 197, 198	mpsyl 68	. . . . . . . 8 |- ( ph -> Disj p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
200	199	adantr 481	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> Disj p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) )
201		measvuni 30277	. . . . . . 7 |- ( ( M e. (measures ` dom M ) /\ A. p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M /\ ( ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ~<_ om /\ Disj p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) ) -> ( M ` U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) = Σ* p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
202	179, 181, 182, 200, 201	syl112anc 1330	. . . . . 6 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` U_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) = Σ* p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
203		ssfi 8180	. . . . . . . . 9 |- ( ( ( ran F X. ran G ) e. Fin /\ ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( ran F X. ran G ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) e. Fin )
204	117, 118, 203	sylancl 694	. . . . . . . 8 |- ( ph -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) e. Fin )
205	204	adantr 481	. . . . . . 7 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) e. Fin )
206		simpll 790	. . . . . . . 8 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ph )
207		simpr 477	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) )
208	118, 207	sseldi 3601	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> p e. ( ran F X. ran G ) )
209		xp1st 7198	. . . . . . . . 9 |- ( p e. ( ran F X. ran G ) -> ( 1st ` p ) e. ran F )
210	208, 209	syl 17	. . . . . . . 8 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( 1st ` p ) e. ran F )
211		xp2nd 7199	. . . . . . . . 9 |- ( p e. ( ran F X. ran G ) -> ( 2nd ` p ) e. ran G )
212	208, 211	syl 17	. . . . . . . 8 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( 2nd ` p ) e. ran G )
213		oveq12 6659	. . . . . . . . . . . . . . . 16 |- ( ( x = .0. /\ y = .0. ) -> ( x .+ y ) = ( .0. .+ .0. ) )
214		sibfof.5	. . . . . . . . . . . . . . . 16 |- ( ph -> ( .0. .+ .0. ) = ( 0g ` K ) )
215	213, 214	sylan9eqr 2678	. . . . . . . . . . . . . . 15 |- ( ( ph /\ ( x = .0. /\ y = .0. ) ) -> ( x .+ y ) = ( 0g ` K ) )
216	215	ex 450	. . . . . . . . . . . . . 14 |- ( ph -> ( ( x = .0. /\ y = .0. ) -> ( x .+ y ) = ( 0g ` K ) ) )
217	216	necon3ad 2807	. . . . . . . . . . . . 13 |- ( ph -> ( ( x .+ y ) =/= ( 0g ` K ) -> -. ( x = .0. /\ y = .0. ) ) )
218		neorian 2888	. . . . . . . . . . . . 13 |- ( ( x =/= .0. \/ y =/= .0. ) <-> -. ( x = .0. /\ y = .0. ) )
219	217, 218	syl6ibr 242	. . . . . . . . . . . 12 |- ( ph -> ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
220	219	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
221	220	ralrimivva 2971	. . . . . . . . . 10 |- ( ph -> A. x e. B A. y e. B ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
222	206, 221	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> A. x e. B A. y e. B ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) )
223	69	a1i 11	. . . . . . . . . . . . 13 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) C_ ( `' .+ " { z } ) )
224	223	sselda 3603	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> p e. ( `' .+ " { z } ) )
225		fniniseg 6338	. . . . . . . . . . . . 13 |- ( .+ Fn ( B X. B ) -> ( p e. ( `' .+ " { z } ) <-> ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) ) )
226	206, 63, 225	3syl 18	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( p e. ( `' .+ " { z } ) <-> ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) ) )
227	224, 226	mpbid 222	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) )
228		simpr 477	. . . . . . . . . . . 12 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> ( .+ ` p ) = z )
229		1st2nd2 7205	. . . . . . . . . . . . . . 15 |- ( p e. ( B X. B ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
230	229	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( p e. ( B X. B ) -> ( .+ ` p ) = ( .+ ` <. ( 1st ` p ) , ( 2nd ` p ) >. ) )
231		df-ov 6653	. . . . . . . . . . . . . 14 |- ( ( 1st ` p ) .+ ( 2nd ` p ) ) = ( .+ ` <. ( 1st ` p ) , ( 2nd ` p ) >. )
232	230, 231	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( p e. ( B X. B ) -> ( .+ ` p ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
233	232	adantr 481	. . . . . . . . . . . 12 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> ( .+ ` p ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
234	228, 233	eqtr3d 2658	. . . . . . . . . . 11 |- ( ( p e. ( B X. B ) /\ ( .+ ` p ) = z ) -> z = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
235	227, 234	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> z = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
236		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) )
237	236	eldifbd 3587	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> -. z e. { ( 0g ` K ) } )
238		velsn 4193	. . . . . . . . . . . 12 |- ( z e. { ( 0g ` K ) } <-> z = ( 0g ` K ) )
239	238	necon3bbii 2841	. . . . . . . . . . 11 |- ( -. z e. { ( 0g ` K ) } <-> z =/= ( 0g ` K ) )
240	237, 239	sylib 208	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> z =/= ( 0g ` K ) )
241	235, 240	eqnetrrd 2862	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( ( 1st ` p ) .+ ( 2nd ` p ) ) =/= ( 0g ` K ) )
242	180, 76	sylanl2 683	. . . . . . . . . . 11 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> p e. ( B X. B ) )
243	242, 89	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( 1st ` p ) e. B )
244	242, 102	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( 2nd ` p ) e. B )
245		oveq1 6657	. . . . . . . . . . . . 13 |- ( x = ( 1st ` p ) -> ( x .+ y ) = ( ( 1st ` p ) .+ y ) )
246	245	neeq1d 2853	. . . . . . . . . . . 12 |- ( x = ( 1st ` p ) -> ( ( x .+ y ) =/= ( 0g ` K ) <-> ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) ) )
247		neeq1 2856	. . . . . . . . . . . . 13 |- ( x = ( 1st ` p ) -> ( x =/= .0. <-> ( 1st ` p ) =/= .0. ) )
248	247	orbi1d 739	. . . . . . . . . . . 12 |- ( x = ( 1st ` p ) -> ( ( x =/= .0. \/ y =/= .0. ) <-> ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) ) )
249	246, 248	imbi12d 334	. . . . . . . . . . 11 |- ( x = ( 1st ` p ) -> ( ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) <-> ( ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) -> ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) ) ) )
250		oveq2 6658	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` p ) -> ( ( 1st ` p ) .+ y ) = ( ( 1st ` p ) .+ ( 2nd ` p ) ) )
251	250	neeq1d 2853	. . . . . . . . . . . 12 |- ( y = ( 2nd ` p ) -> ( ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) <-> ( ( 1st ` p ) .+ ( 2nd ` p ) ) =/= ( 0g ` K ) ) )
252		neeq1 2856	. . . . . . . . . . . . 13 |- ( y = ( 2nd ` p ) -> ( y =/= .0. <-> ( 2nd ` p ) =/= .0. ) )
253	252	orbi2d 738	. . . . . . . . . . . 12 |- ( y = ( 2nd ` p ) -> ( ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) <-> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) ) )
254	251, 253	imbi12d 334	. . . . . . . . . . 11 |- ( y = ( 2nd ` p ) -> ( ( ( ( 1st ` p ) .+ y ) =/= ( 0g ` K ) -> ( ( 1st ` p ) =/= .0. \/ y =/= .0. ) ) <-> ( ( ( 1st ` p ) .+ ( 2nd ` p ) ) =/= ( 0g ` K ) -> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) ) ) )
255	249, 254	rspc2v 3322	. . . . . . . . . 10 |- ( ( ( 1st ` p ) e. B /\ ( 2nd ` p ) e. B ) -> ( A. x e. B A. y e. B ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) -> ( ( ( 1st ` p ) .+ ( 2nd ` p ) ) =/= ( 0g ` K ) -> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) ) ) )
256	243, 244, 255	syl2anc 693	. . . . . . . . 9 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( A. x e. B A. y e. B ( ( x .+ y ) =/= ( 0g ` K ) -> ( x =/= .0. \/ y =/= .0. ) ) -> ( ( ( 1st ` p ) .+ ( 2nd ` p ) ) =/= ( 0g ` K ) -> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) ) ) )
257	222, 241, 256	mp2d 49	. . . . . . . 8 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) )
258	3, 4, 11, 12, 13, 14, 15, 16, 17, 19, 2, 78	sibfinima 30401	. . . . . . . 8 |- ( ( ( ph /\ ( 1st ` p ) e. ran F /\ ( 2nd ` p ) e. ran G ) /\ ( ( 1st ` p ) =/= .0. \/ ( 2nd ` p ) =/= .0. ) ) -> ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) e. ( 0 [,) +oo ) )
259	206, 210, 212, 257, 258	syl31anc 1329	. . . . . . 7 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) e. ( 0 [,) +oo ) )
260	205, 259	esumpfinval 30137	. . . . . 6 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> Σ* p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) = sum_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
261	176, 202, 260	3eqtrd 2660	. . . . 5 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) = sum_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
262		rge0ssre 12280	. . . . . . 7 |- ( 0 [,) +oo ) C_ RR
263	262, 259	sseldi 3601	. . . . . 6 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) e. RR )
264	205, 263	fsumrecl 14465	. . . . 5 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> sum_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) e. RR )
265	261, 264	eqeltrd 2701	. . . 4 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. RR )
266	179	adantr 481	. . . . . . 7 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> M e. (measures ` dom M ) )
267	180, 112	sylanl2 683	. . . . . . 7 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M )
268		measge0 30270	. . . . . . 7 |- ( ( M e. (measures ` dom M ) /\ ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) e. dom M ) -> 0 <_ ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
269	266, 267, 268	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) /\ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ) -> 0 <_ ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
270	205, 263, 269	fsumge0 14527	. . . . 5 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> 0 <_ sum_ p e. ( ( `' .+ " { z } ) i^i ( ran F X. ran G ) ) ( M ` ( ( `' F " { ( 1st ` p ) } ) i^i ( `' G " { ( 2nd ` p ) } ) ) ) )
271	270, 261	breqtrrd 4681	. . . 4 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> 0 <_ ( M ` ( `' ( F oF .+ G ) " { z } ) ) )
272		elrege0 12278	. . . 4 |- ( ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) <-> ( ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. RR /\ 0 <_ ( M ` ( `' ( F oF .+ G ) " { z } ) ) ) )
273	265, 271, 272	sylanbrc 698	. . 3 |- ( ( ph /\ z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ) -> ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) )
274	273	ralrimiva 2966	. 2 |- ( ph -> A. z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) )
275		eqid 2622	. . 3 |- (sigaGen ` ( TopOpen ` K ) ) = (sigaGen ` ( TopOpen ` K ) )
276		eqid 2622	. . 3 |- ( 0g ` K ) = ( 0g ` K )
277		eqid 2622	. . 3 |- ( .s ` K ) = ( .s ` K )
278		eqid 2622	. . 3 |- (RRHom ` (Scalar ` K ) ) = (RRHom ` (Scalar ` K ) )
279	27, 28, 275, 276, 277, 278, 26, 16	issibf 30395	. 2 |- ( ph -> ( ( F oF .+ G ) e. dom ( Ksitg M ) <-> ( ( F oF .+ G ) e. ( dom MMblFnM (sigaGen ` ( TopOpen ` K ) ) ) /\ ran ( F oF .+ G ) e. Fin /\ A. z e. ( ran ( F oF .+ G ) \ { ( 0g ` K ) } ) ( M ` ( `' ( F oF .+ G ) " { z } ) ) e. ( 0 [,) +oo ) ) ) )
280	174, 142, 274, 279	mpbir3and 1245	1 |- ( ph -> ( F oF .+ G ) e. dom ( Ksitg M ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U.cuni 4436   U_ciun 4520  Disj wdisj 4620   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   omcom 7065   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   ~<_ cdom 7953   ~< csdm 7954   Fincfn 7955   RRcr 9935   0cc0 9936   +oocpnf 10071   <_ cle 10075   [,)cico 12177   sum_csu 14416   Basecbs 15857  Scalarcsca 15944   .scvsca 15945   TopOpenctopn 16082   0gc0g 16100   Topctop 20698   TopSpctps 20736   Clsdccld 20820   Frect1 21111  RRHomcrrh 30037  Σ^*cesum 30089  sigAlgebracsiga 30170  sigaGencsigagen 30201  measurescmeas 30258  MblFnMcmbfm 30312  sitgcsitg 30391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-ordt 16161  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-ps 17200  df-tsr 17201  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-ring 18549  df-cring 18550  df-subrg 18778  df-abv 18817  df-lmod 18865  df-scaf 18866  df-sra 19172  df-rgmod 19173  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-t1 21118  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-tmd 21876  df-tgp 21877  df-tsms 21930  df-trg 21963  df-xms 22125  df-ms 22126  df-tms 22127  df-nm 22387  df-ngp 22388  df-nrg 22390  df-nlm 22391  df-ii 22680  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-esum 30090  df-siga 30171  df-sigagen 30202  df-meas 30259  df-mbfm 30313  df-sitg 30392

This theorem is referenced by:  sitmcl  30413